Why does AlphaGo Zero select move based on exponentiated visit count?
The answer is surprisingly hidden in the original AlphaGo paper:


  At the end of search AlphaGo selects the action with maximum visit count; this
  is less sensitive to outliers than maximizing action value.


Unfortunately, there did not appear to be further details in the paper or in the related reference. The root child node (corresponding to an action) with maximum visit count is fittingly known as the robust child, as described here and referenced in a MCTS survey here.